MinT - Best Known (69, 69+37, s)-Nets in Base 3

Best Known (69, 69+37, s)-Nets in Base 3

(69, 69+37, 128)-Net over F₃ — Constructive and digital

Digital (69, 106, 128)-net over F₃, using

6 times m-reduction [i] based on digital (69, 112, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 56, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(69, 69+37, 165)-Net over F₃ — Digital

Digital (69, 106, 165)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+37, 2275)-Net in Base 3 — Upper bound on s

There is no (69, 106, 2276)-net in base 3, because

1 times m-reduction [i] would yield (69, 105, 2276)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 126 178844 922553 160982 895804 564127 997696 628851 193209 > 3¹⁰⁵ [i]